Abstract

We prove a series of new results for the irrationality measure of some values of $\arctan <!--FUNCTION\; APPLICATION-->x$. Some time ago we applied symmetric complex integrals to approximate values of the form $\arctan <!--FUNCTION\; APPLICATION-->(1∕n)$, where $n\in \mathbb{N}$ is a natural number. This method gave new estimates for the numbers $\arctan <!--FUNCTION\; APPLICATION-->\frac{1}{2}$ and $\arctan <!--FUNCTION\; APPLICATION-->\frac{1}{3}$ only. To deal with some other values of $\arctan <!--FUNCTION\; APPLICATION-->x$ we modify the main construction. In the present paper, we consider a new integral which is based on an idea of Q. Wu and does not have a property of symmetry of the integrand. Integral construction of such a type allows us to improve estimates for the irrationality measure of some values of $\arctan <!--FUNCTION\; APPLICATION-->(s∕r)$ for some natural $s,r\in \mathbb{N}$, $s<r$.

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